We define and study the complexity of robust polynomials for Boolean functions and the related fault-tolerant quantum decision trees, where input bits are perturbed by noise. We show that, in contrast to the classical model of Feige et. al., every Boolean function can be computed by O(n) quantum queries even in the model with noise. This implies, for instance, the somewhat surprising result that every Boolean function has robust degree bounded by O(n).
This joint work with Ilan Newman, Hein Roehrig, and Ronald de Wolf